Page 338 - Cam Design Handbook

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326 CAM DESIGN HANDBOOK

extremely good reason to do this in terms of the kinetic energy. The coordinates of the
mass center are described by

1 1 1 1
r
r
x CR = Ú xdm = Ú xdV y = Ú ydm = Ú ydV (11.23)
cg
m m m m
rigid body rigid body rigid body rigid body
for any coordinate system (x,y) in the plane of motion. If the object has constant density,
this can be expressed solely in terms of the volume V
1 1
x = Ú xdV y = Ú ydV (11.24)
cg cg
V V
rigid body rigid body
in which case the mass center is said to coincide with the volume centroid of the body.
Rearranging Eq. 11.23,
cg )
Ú ( x - x ) dm = Ú ( y y dm = 0 (11.25)
-
cg
rigid body rigid body
from which it follows that
( È x - x )˘
Ú r dm = Ú Í ( cg ˙ dm = 0 (11.26)
cg
-
cg
rigid body rigid body Î yy ) ˚
where the vector r cg represents the vector from the center of mass to an arbitrary point on
the body. This relationship, in turn, enables the kinetic energy for the general case of trans-
lation and rotation in a plane (Fig. 11.5) to be simplified by splitting it into three terms:

V = V + w ¥ r cg
cg
w ¥ r
V cg
w cg
w ¥ r cg
V cg
r cg

FIGURE 11.5. Rigid body planar motion: translation with rotation.